Question

I was stuck on the following review problem on hypothesis testing. Could you please guide me on how to approach this type of problem?

Stacy, a professional soccer player, had a field goal percentage (i.e., probability of making a shot) of p₀ = 60% before needing to take a season off to recover from an injury.

(a) Since returning to the game from injury, Stacy has made 13 out of n = 20 shots. Is Stacy's new, post-injury field goal percentage higher than her old percentage p₀? Perform a suitable one-sided hypothesis test and state your conclusion, taking α = 0.05.

(b) Suppose that the true new field goal percentage is p, where p ∈ (0.6, 1). If we perform a one-sided test as above and want to achieve a type-I error rate of 0.05 and type-II error rate of 0.025, what is the number of shots n needed since returning from injury? Provide an approximate formula as a function of p, and compute the values of n for each of p = 0.8, 0.7, 0.61 (Notice that if p is very close to 0.6 then you may need a very large number of shots.)

(c) Stacy scored X points in a high school game. Knowing that X is greater than 100, find X.

Answer 1

a) We have to test,

H0 : p = p0 = 0.6(given) against, H1 : p > 0.6(p0)

After returning from injury, Stacy has made 13 out of n = 20 shots i.e. observed percentage, = 13/20 = 0.65

Now, P(rejecting H0 when H0 is true) = specified significance level = 0.05

under H0, X = no. of goals Stacy made, ~ Binomial(20, 0.6) and Xc be the critical value i.e. we reject H0 if X > Xc.

From the cumulative probability table of Binomial distn., For n=20,

Since, the observed no. of goals, 13

Since, the observed no. of goals, 13 15, there are not enough evidance to reject the null hypothesis at 0.05% level of significance.

b) Z-test:

Test statistic:

, p is the sample proportion, where p ∈ (0.6, 1), n is the sample size.

Given, alpha level is 0.05.

type-II erroe is 0.025

For p = 0.8, 16 < n < 23

For p = 0.7, 64 < n < 92

For p = 0.61, 6494 < n < 9219

=> if p is very close to 0.6 then we may need a very large number of shots.